Adaptive integration for Capon-type beamformers by exploring Riemannian geometry on covariance matrices

Abstract

Adaptive, or Capon beamformers, offers several advantages compared to standard alternatives in underwater sensing. Furthermore, for hull mounted sensor arrays, adaptive beamformers is a convenient way of mitigating platform induced interference. The underlying difficulty in implementing a data-adaptive approach is how to obtain an accurate estimate of the data covariance matrix, R. The combination of high-dimensionality and time-variation renders significant challenges in maintaining accurate covariance estimates. A common way to approach this issue is to update the covariance matrix using a weighted Euclidean average, Rnew = a Rold + (1-a) Rinst, where Rinst is some instantaneous measurement. Meanwhile, the time-variation naturally form a trajectory on the covariance matrix manifold. With this in mind, it makes much more sense to smooth the trajectory on the space of covariance matrices using proper metrics. The purpose of this paper is to take the geometry of covariance matrices into account while performing adequate covariance matrix estimation. Specifically, we propose a simple and elegant algorithm to track the covariance based on an update through means on Riemannian manifolds. This solution is then incorporated into an adaptive beamformer, and initial evaluations based on both simulated and real data show slight improvements compared to standard approaches.